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Evaluate Problems using fluid dynamics

  1. Jan 14, 2017 #1
    I have a diagram similar to the following. Water entering the larger end is at 20degreesC.

    1. The larger end has a diameter of 8cm and Area 50.26cm2.
    2. The small side has a diameter of 3cm and Area 7.0685cm2.
    3. The water jet exerts a force of 87N on a flat plate at an unknown distance.
    4. Assuming no frictions and a steady flow
    5. Liquid is water
    Calculate the velocity at nozzle exit.


    VenturiFlow.png




    2. Relevant equations

    + ∗∗+ /∗∗ = t

    p1 - p2 = density/2 (v2 ^2 - v1^2)
    https://wikimedia.org/api/rest_v1/media/math/render/svg/5867d9ef7ba2631627c12a000ec0096b9550c390

    P = F/A

    A1V1 = A2V2

    3. The attempt at a solution

    I have calculated the pressure at the smaller end using P=F/A . P2 = 12..3N/cm2

    Using A1V1 = A2V2 I know that V2 = 7.11 * V1

    I have inputted my values in the Bernoulli Equation p1 - p2 = density/2 (v2 ^2 - v1^2) , however I am ending up with both velocities as unknowns and also P1 is unknown.

    I am getting stuck when I come to calculate Pressure at point 1 and Velocity at Point 2.

    Any pointers or suggestions would be appreciated.
     
    Last edited: Jan 14, 2017
  2. jcsd
  3. Jan 14, 2017 #2
    Are you currently learning about macroscopic momentum balances by any chance?
     
  4. Jan 14, 2017 #3
    No, we have covered the Bernoulli equation, continuity of flow, the Pitot tube and the Venturi Meter.
     
  5. Jan 14, 2017 #4
    It the smaller end open to the air?
     
  6. Jan 14, 2017 #5
    Yes it is. I will try to update the picture of my question paper.
     

    Attached Files:

  7. Jan 14, 2017 #6
    So, if the exit pressure is 0 Pa gauge, how do you interpret "The water jet exerts a force of 87N at the smaller end."?
     
  8. Jan 14, 2017 #7
    I must have falsely interpreted my question. That was one of my mistakes that I am realising now. The 87 N is exerted on the flat plate which is a completely different story. I will update my question. I apologise for misinterpreting the question.
     
  9. Jan 14, 2017 #8
    No big deal.
     
  10. Jan 14, 2017 #9
    the force is due to the pressure of moving fluid there
    F=P*A
     
  11. Jan 14, 2017 #10
    I still cannot use the continuity of flow theorem A1/A2 = V1/V2 or Bernoulli's equation as I do not know any velocity or Pressure at the wide end.
     
  12. Jan 14, 2017 #11
    The OP already stated that the exit pressure is zero.
     
  13. Jan 14, 2017 #12
    The difficulty is interpreting what they mean by "The water jet exerts a force of 87N at the smaller end." I would interpret this as the force that the flowing fluid exerts on the body of the nozzle. But, if this is the correct interpretation, then to get what you want will require using a momentum balance on the flow in addition to the Bernoulli equation.
     
  14. Jan 14, 2017 #13
    pressure on the flat plate where water hits,F/A2=p2+density/2(v2^2) (when P by air=0(considering plate near to exit))
     
  15. Jan 14, 2017 #14
    This is certainly also a reasonable interpretation, if it is in agreement with the actual problem geometry.
     
  16. Jan 14, 2017 #15

    TSny

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    Gold Member

    For part (a) I don't think the Bernoulli Eq plays a role. The force on the plate is related to the rate of loss of momentum of the stream of water as it hits the plate.
     
  17. Jan 14, 2017 #16
    So I end up with : 12.3 = p2 + 1000/2(V2^2) .

    Should P2 be taken as 0 in this case since it is at atmospheric pressure ?
     
  18. Jan 14, 2017 #17
    I came across this video while doing research however it does not match with the formula I was given earlier on.

    Can anyone confirm if this video is correct ?
     
  19. Jan 14, 2017 #18

    TSny

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    The video derives the correct expression for the force on the plate in terms of ##\rho##, ##v## and the cross-sectional area ##A## of the stream.

    (At the end of the video (at time 9:40), he forgot to divide the change in momentum by the time in order to express the force.)
     
  20. Jan 14, 2017 #19
    Thanks for your insight on this matter. This means that ,F/A2=p2+density/2(v2^2) (given earlier) is incorrect ? Or am I still missing something here?
     
  21. Jan 14, 2017 #20
    The equation I get using a momentum balance is $$F=\rho v^2 A$$
     
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